Strictly speaking, the pronunciation of these two words is quite different, but it is a common error to get “formerly” and “formally” mixed up both in speech and in writing. However, their definitions are also entirely different. If you swap one for the other, you can completely alter the meaning of the sentence.
Formerly refers to something that happened in the past. This is inherently related to the word former, which would also denote something that happened previously. We could say, for example, that Hong Kong was formerly a British colony. This was something that was once true, but Hong Kong is no longer a part of the British empire.
Similarly, I could say that I was formerly employed by a movie theatre where I popped popcorn and sold tickets. This is to say that I worked at the movie theatre at some point in the past, but I am not working there anymore.
Formally, on the other hand, is based on the root word “formal.” This could refer to something that was performed in an official manner according to the standards of convention for those circumstances. For instance, the Colony of British Columbia was formally established in 1858. This important event made the establishment of BC official. A person can be formally introduced too.
Formal could also refer to formal wear (or formal dress), which is clothing that is suitable for a formal social event like a wedding. You would be told to dress formally, as opposed to dressing casually, for such events.
Switching one for the other, as mentioned, can completely change the meaning of the sentence.
- John Smith, formerly the head of XYZ Company, has declared bankruptcy.
- John Smith, formally the head of XYZ Company, has declared bankruptcy.
With the first sentence, we learn that John Smith was once the head of the company, but he is no longer. With the second sentence, we learn that John Smith is the current head of the company in some official manner, likely as President or CEO.
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Don’t forget, formally can also mean “Following or being in accord with accepted forms” (definition from The Free Dictionary), as in the sentence “He proved it formally but gave his students no intuitive feel for the matter” (courtesy of Wiktionary).
Same definition as what Michael gives just worded differently. “Official Manner” and Standards of Convention” = “In accord with accepted forms”.